﻿Elements of Geometry. 17 



very similar to those that he employs, of the propositions 

 which he actually states. References are throughout to 

 Todhunter's edition. 



(2) But two preliminary questions must be asked. F-irst, 

 can an experimental science be deductive at all? Certainly 

 it can. A deduction from a law is an application of that 

 law in particular circumstances which were not examined 

 when it was formulated. If, after examining the sides of 

 squares and of triangles, I assert the general law that 

 all straight lines have measurable lengths, and then, without 

 further experiment, assert that the diagonals of squares, 

 which are also straight lines, are also measurable, I am 

 miking a deduction. It may be true that there is some- 

 thing precarious about the results of such deduction — 

 that question is not raised here, — but the deduction itself 

 is quite unexceptionable ; the falsity of the conclusion is 

 definitely inconsistent with the truth of the premises. 

 If deubt is raised concerning the conclusion, the ultimate 

 means of resolving it is by experiment ; but experimental 

 science, in the hands of its greatest exponents, consists in 

 asserting- such general laws that doubt does not arise 

 concerning the results of deduction based on them. 



The second question is whether there are truly laws 

 which make measurement possible. The question is dis- 

 cussed at length in my ' Physics/ Part II., the results and 

 nomenclature of which will be used freely in what follows. 

 But there is one matter which may receive special mention 

 here, because it is concerned with " incommensurables," 

 which are often (but falsely) believed to be of especial 

 importance in geometry. Measurement is possible when, 

 by means of definitions of equality and addition, a standard 

 series of the property in question can be established, starting 

 from some arbitrary unit, such that any system having the 

 property is equal in respect of it to some one member of 

 the standard series. Now (it might be argued) such mea- 

 surement is not possible for length, because the diagonal of 

 a square cannot be equal to any member of a standard series 

 based on the side as unit ; indeed that result is actually 

 proved by Euclid. Consequently it is patently absurd to 

 pretend that Euclid's propositions can be derived from an 

 assumption, namely that measurement is possible, which is 

 inconsistent with its conclusions. 



One method of escape from this difficulty may be 

 mentioned, although it will not be adopted. A slight 



Phil. Mag. S. 6. Vol. U. No. 259. July 1922. 



